The idea of the proofs is inspired from [1]}, but the details and techniques are wholly different.
The proof starts by identifying a weakly interacting particle system for which the LDP is known, and that is exponentially close (see Definition REF ) to the state processes, then use the closeness property to "transfer" the LDP to the state processes.
Since the state processes of the \(N\)  agents at equilibrium are characterized by a system of forward-backward SDEs, the identification of the suitable auxiliary particle system whose LDP is known uses the well-known technique of decoupling fields.
However, the proper decoupling field turns out to be the solution (when it exists) of the master equation.
The main difficulty lies in the proof of the exponential closeness property.
This is based on a priori bounds for systems of FBSDEs at least provided that the solution of the master equation is Lipschitz–continuous.
In addition to the fact that we consider games allowing control interaction and open-loop controls in the non-uniqueness MFG case, the essential difference with [1]} is the assumptions made on the PDEs.
On the one hand, we do not make use of the \(N\) -player PDE, and on the other hand we require the solution of the master equation to be Lipschitz–continuous; we do not make assumptions pertaining to its second derivative.
This is an interesting by-product of the convergence method of [3]} which is based on Pontryagin's stochastic maximum principle.
The (very) rough intuition for this gain of smoothness is that in the present case, the FBSDE (resp. the master equation) allows to represent the control (or the "derivative of the value function").
In contrast, the Hamilton-Jacobi-Bellman equation used in [1]} represents the value function itself.
